Rotating-wave approximation: Difference between revisions

Revision as of 10:45, 15 May 2007

Latest revision as of 18:36, 13 August 2025

(69 intermediate revisions by 49 users not shown)

Line 1:

{{Short description|Model used in atom optics and magnetic resonance}}

The '''rotating wave approximation''' is an approximation used in [[atom optics]] and [[magnetic resonance]]. In this approximation, terms in a Hamiltonian which oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic resonance, and the intensity is low. Explicitly, terms in the Hamiltonians which oscillate with frequencies <math>\omega_L+\omega_0 </math> are neglected, while terms which oscillate with frequencies <math>\omega_L-\omega_0 </math> are kept, where <math> \omega_L </math> is the light frequency and <math> \omega_0</math> is a transition frequency.

{{Refimprove|date=August 2013}}

The '''rotating-wave approximation''' is an approximation used in [[atom optics]] and [[Nuclear magnetic resonance|magnetic resonance]]. In this approximation, terms in a [[Hamiltonian (quantum mechanics)|Hamiltonian]] that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic transition, and the intensity is low.<ref name="WuYang2007">{{cite journal |last1=Wu |first1=Ying |last2=Yang |first2=Xiaoxue |title=Strong-Coupling Theory of Periodically Driven Two-Level Systems |journal=Physical Review Letters |volume=98 |issue=1 |year=2007 |issn=0031-9007 |doi=10.1103/PhysRevLett.98.013601 |bibcode=2007PhRvL..98a3601W |pmid=17358474 |page=013601}}</ref> Explicitly, terms in the Hamiltonians that oscillate with frequencies <math>\omega_L + \omega_0</math> are neglected, while terms that oscillate with frequencies <math>\omega_L - \omega_0</math> are kept, where <math>\omega_L</math> is the light frequency, and <math>\omega_0</math> is a transition frequency.

The name of the approximation stems from the form of the Hamiltonian in the [[interaction picture]], as shown below. By switching to this picture the evolution of an atom due to the corresponding atomic Hamiltonian is absorbed into the system [[bra-ket notation|ket]], leaving only the evolution due to the interaction of the atom with the light field to consider. It is in this picture that the rapidly-oscillating terms mentioned previously can be neglected. Since in some sense the interaction picture can be thought of as rotating with the system ket only that part of the electromagnetic wave that approximately co-rotates is kept; the counter-rotating component is discarded.

The name of the approximation stems from the form of the Hamiltonian in the [[interaction picture]], as shown below. By switching to this picture the evolution of an atom due to the corresponding atomic Hamiltonian is absorbed into the system [[bra–ket notation|ket]], leaving only the evolution due to the interaction of the atom with the light field to consider. It is in this picture that the rapidly oscillating terms mentioned previously can be neglected. Since in some sense the interaction picture can be thought of as rotating with the system ket only that part of the electromagnetic wave that approximately co-rotates is kept; the counter-rotating component is discarded.

==Mathematical formulation==

For simplicity consider a [[two-state quantum system|two-level atomic system]] with [[excited state|excited]] and ground states <math>|\text{e}\rangle</math> and <math>|\text{g}\rangle</math> respectively (using the [[bra-ket notation|Dirac bracket notation]]). Let the energy difference between the states be <math>\hbar\omega_0</math> so that <math>\omega_0</math> is the transition frequency of the system. Then the unperturbed [[Hamiltonian]] of the atom can be written as

The rotating-wave approximation is closely related to, but different from, the [[Redfield_equation#Secular_approximation|secular approximation]].<ref>{{cite journal |first1=H. |last1=Mäkelä |first2=M. |last2=Möttönen |title=Effects of the rotating-wave and secular approximations on non-Markovianity |url=https://link.aps.org/doi/10.1103/PhysRevA.88.052111 |journal=Physical Review A |date=13 November 2013 |pages=052111 |volume=88 |issue=5| doi=10.1103/PhysRevA.88.052111|arxiv=1306.6301 |bibcode=2013PhRvA..88e2111M }}</ref>

<math>H_0=\hbar\omega_0|\text{e}\rangle\langle\text{e}|</math>

== Mathematical formulation ==

Suppose the atom is placed at <math>z=0</math> in an external (classical) [[electric field]] of frequency <math>\omega_L</math>, given by <math>\vec{E}(z,t)=\vec{E}_0(z)e^{-i\omega_Lt}+\vec{E}_0^*(z)e^{i\omega_Lt}</math> (so that the field contains both positive- and negative-frequency modes in general). Then under the [[dipole approximation]] the [[interaction Hamiltonian]] can be expressed as

For simplicity consider a [[two-state quantum system|two-level atomic system]] with [[ground state|ground]] and [[excited state|excited]] states <math>|\text{g}\rangle</math> and <math>|\text{e}\rangle</math>, respectively (using the [[bra–ket notation|Dirac bracket notation]]). Let the energy difference between the states be <math>\hbar\omega_0</math> so that <math>\omega_0</math> is the transition frequency of the system. Then the unperturbed [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the atom can be written as

: <math>H_0 = \frac{\hbar\omega_0}{2}|\text{e}\rangle\langle\text{e}|-\frac{\hbar\omega_0}{2}|\text{g}\rangle\langle\text{g}|</math>.

<math>H_I=-\vec{d}\cdot\vec{E}</math>

Suppose the atom experiences an external classical [[electric field]] of frequency <math>\omega_L</math>, given by

where <math>\vec{d}</math> is the [[transition dipole moment|dipole moment operator]] of the atom. The total Hamiltonian for the atom-light system is therefore <math>H=H_0+H_I.</math> The atom does not have a dipole moment when it is in an [[energy eigenstate]], so <math>\langle\text{e}|\vec{d}|\text{e}\rangle=\langle\text{g}|\vec{d}|\text{g}\rangle=0.</math> This means that defining <math>\vec{d}_{\text{eg}}:=\langle\text{e}|\vec{d}|\text{g}\rangle</math> allows the dipole operator to be written as

<math>\vec{E}(t) = \vec{E}_0 e^{-i\omega_Lt} +\vec{E}_0^* e^{i\omega_Lt}</math>; e.g., a [[plane wave]] propagating in space. Then under the [[dipole#Torque on a dipole|dipole approximation]] the interaction Hamiltonian between the atom and the electric field can be expressed as

: <math>\vec{d}H_1 = -\vec{d}_{ \text{eg}}|cdot \text{e}\rangle\langle\text{g}|+\textvec{H.c.E}</math>,

where <math>\vec{d}</math> is the [[transition dipole moment|dipole moment operator]] of the atom. The total Hamiltonian for the atom-light system is therefore <math>H = H_0 + H_1.</math> The atom does not have a dipole moment when it is in an [[energy eigenstate]], so <math>\left\langle\text{e}\left|\vec{d}\right|\text{e}\right\rangle = \left\langle\text{g}\left|\vec{d}\right|\text{g}\right\rangle = 0.</math> This means that defining <math>\vec{d}_\text{eg} \mathrel{:=} \left\langle\text{e}\left|\vec{d}\right|\text{g}\right\rangle</math> allows the dipole operator to be written as

(with `H.c.' denoting the [[Hermitean conjugate]]). The interaction Hamiltonian can then be shown to be (see the Derivations section below)

: <math>H_I\vec{d} =- \hbarvec{d}_\left(text{eg}|\Omega e^text{-ie}\omega_Ltrangle\langle\text{g}| + \tildevec{\Omegad}e^{i_\omega_Lttext{eg}\right)^*|\text{eg}\rangle\langle\text{ge}|</math>

-\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|</math>

(with <math>^*</math> denoting the [[complex conjugate]]). The [[#Derivation|interaction Hamiltonian can then be shown to be]]

where <math>\Omega</math> is the [[Rabi frequency]] and <math>\tilde{\Omega}:=\hbar^{-1}\vec{d}_\text{eg}\cdot\vec{E}_0^*</math> is the counter-rotating frequency. To see why the <math>\tilde{\Omega}</math> terms are called `counter-rotating' consider a [[unitary transformation]] to the interaction or Dirac picture where the transformed Hamiltonian <math>\bar{H}</math> is given by

: <math>H_1 =

<math>\bar{H}=-\hbar\left(\Omega e^{-i\Delta t}+\tilde{\Omega}e^{i(\omega_L+\omega_0)t}\right)|\text{e}\rangle\langle\text{g}|

-\hbar\left(\tilde{\Omega}^* e^{-i(\omega_L+\omega_0)tomega_Lt} + \tilde{\Omega^*}e^{i\Delta tomega_Lt}\right)|\text{ge}\rangle\langle\text{eg}|,</math>

-\hbar\left(\tilde{\Omega}^* e^{-i\omega_Lt} + \Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|

</math>

where <math>\Omega = \hbar^{-1}\vec{d}_\text{eg} \cdot \vec{E}_0</math> is the [[Rabi frequency]] and <math>\tilde{\Omega} \mathrel{:=} \hbar^{-1}\vec{d}_\text{eg} \cdot \vec{E}_0^*</math> is the counter-rotating frequency. To see why the <math>\tilde{\Omega}</math> terms are called ''counter-rotating'' consider a [[unitary transformation]] to the [[interaction picture|interaction or Dirac picture]] where the transformed Hamiltonian <math>H_{1,I}</math> is given by

where <math>\Delta:=\omega_L-\omega_0</math> is the detuning of the light field.

: <math>H_{1,I} =

===Making the approximation===

-\hbar\left(\Omega e^{-i\Delta \omega t} + \tilde{\Omega}e^{i(\omega_L + \omega_0)t}\right)|\text{e}\rangle\langle\text{g}|

This is the point at which the rotating wave approximation is made. The dipole approximation has been assumed, and for this to remain valid the electric field must be near [[resonance]] with the atomic transition. This means that <math>\Delta\ll\omega_L+\omega_0</math> and the complex exponentials multiplying <math>\tilde{\Omega}</math> and <math>\tilde{\Omega}^*</math> can be considered to be rapidly oscillating. Hence on any appreciable time scale the oscillations will quickly average to 0. The rotating wave approximation is thus the claim that these terms are negligible and the Hamiltonian can be written in the interaction picture as

-\hbar\left(\tilde{\Omega}^* e^{-i(\omega_L + \omega_0)t} + \Omega^* e^{i\Delta \omega t}\right)|\text{g}\rangle\langle\text{e}|,

</math>

where <math>\Delta \omega \mathrel{:=} \omega_L - \omega_0</math> is the detuning between the light field and the atom.

<math>\bar{H}_\text{RWA}=-\hbar\Omega e^{-i\Delta t}|\text{e}\rangle\langle\text{g}|

-\hbar\Omega^*e^{i\Delta t}|\text{g}\rangle\langle\text{e}|.</math>

=== Making the approximation ===

Finally, in the [[Schrödinger picture]] the Hamiltonian is given by

[[File:TLSRWA.gif|thumb|Two-level-system on resonance with a driving field with (blue) and without (green) applying the rotating-wave approximation.]]

This is the point at which the rotating wave approximation is made. The dipole approximation has been assumed, and for this to remain valid the electric field must be near [[resonance]] with the atomic transition. This means that <math>\Delta \omega \ll \omega_L + \omega_0</math> and the complex exponentials multiplying <math>\tilde{\Omega}</math> and <math>\tilde{\Omega}^*</math> can be considered to be rapidly oscillating. Hence on any appreciable time scale, the oscillations will quickly average to 0. The rotating wave approximation is thus the claim that these terms may be neglected and thus the Hamiltonian can be written in the interaction picture as

<math>

H_\text{RWA}=\hbar\omega_0|\text{e}\rangle\langle\text{e}|

: <math>H_{1,I}^{\text{RWA}} =

-\hbar\Omega e^{-i\omega_Lt}|\text{e}\rangle\langle\text{g}|

-\hbar\Omega^* e^{-i\omega_LtDelta \omega t}|\text{ge}\rangle\langle\text{eg}|.

-\hbar\Omega^* e^{i\Delta \omega t}|\text{g}\rangle\langle\text{e}|.

</math>

Finally, transforming back into the [[Schrödinger picture]], the Hamiltonian is given by

:<math>H^\text{RWA} =

\frac{\hbar\omega_0}{2}|\text{e}\rangle\langle\text{e}|

- \frac{\hbar\omega_0}{2}|\text{g}\rangle\langle\text{g}|

- \hbar\Omega e^{-i\omega_Lt}|\text{e}\rangle\langle\text{g}|

- \hbar\Omega^* e^{i\omega_Lt}|\text{g}\rangle\langle\text{e}|.

</math>

Another criterion for rotating wave approximation is the weak coupling condition, that is, the Rabi frequency should be much less than the transition frequency.<ref name="WuYang2007"/>

At this point the rotating wave approximation is complete. A common first step beyond this is to remove the remaining time dependence in the Hamiltonian via another unitary transformation.

== Derivation ==

==Derivations==

Given the above definitions the interaction Hamiltonian is

: <math>\begin{align}

H_I & H_1 = -\vec{d}\cdot\vec{E} \\

&= -\left(\vec{d}_\text{eg}|\text{e}\rangle\langle\text{g}| + \vec{d}_\text{eg}^*|\text{g}\rangle\langle\text{e}|\right)

\cdot \left(\vec{E}_0e_0 e^{-i\omega_Lt} + \vec{E}_0^* e^{i\omega_Lt}\right) \\

&= -\left(\vec{d}_\text{eg} \cdot \vec{E}_0e_0 e^{-i\omega_Lt}

+\vec{d}_\text{eg} \cdot \vec{E}_0^* e^{i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|

-\left(\vec{d}_\text{eg}^* \cdot \vec{E}_0e_0 e^{-i\omega_Lt}

+\vec{d}_\text{eg}^* \cdot \vec{E}_0^* e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}| \\

&= -\hbar\left(\Omega e^{-i\omega_Lt} + \tilde{\Omega} e^{i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|

-\hbar\left(\tilde{\Omega}^* e^{-i\omega_Lt} + \Omega^* e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|,

\end{align}</math>

as stated. The next stagestep is to find the Hamiltonian in the [[interaction picture]], <math>\barH_{H1,I}.</math> . The unitary operator required for theunitary transformation is:

<math>U=e^{iH_0t/\hbar},</math>

and an arbitrary state <math>|\psi\rangle</math> transforms to <math>|\bar{\psi}\rangle=U|\psi\rangle.</math> The [[Schrödinger equation]] must still hold in this new picture, so

<math>

\begin{align}

\bar{H}|\bar{\psi}\rangle

U & = e^{iH_0t/\hbar} \\

=i\hbar\partial_t|\bar{\psi}\rangle

& = e^{i \omega_0 t/2 (|\text{e}\rangle \langle\text{e}| - |\text{g}\rangle \langle\text{g}|)} \\

=i\hbar\dot{U}|\psi\rangle+Ui\hbar\partial_t|\psi\rangle

& = \cos\left(i\hbarfrac{\dotomega_0 t}{U2}+UH\right)|\psi\rangle

\left(|\text{e}\rangle \langle\text{e}| + |\text{g}\rangle \langle\text{g}|\right) + i \sin\left(\frac{\omega_0 t}{2}\right) \left(|\text{e}\rangle \langle\text{e}| - |\text{g}\rangle \langle\text{g}|\right) \\

=\left(i\hbar\dot{U}U^\dagger+UHU^\dagger\right)|\bar{\psi}\rangle,

& = e^{-i\omega_0 t/2}|\text{g}\rangle \langle\text{g}| + e^{i \omega_0 t/2} |\text{e}\rangle \langle\text{e}| \\

& = e^{-i\omega_0 t/2}\left(|\text{g}\rangle \langle\text{g}| + e^{i \omega_0 t} |\text{e}\rangle \langle\text{e}|\right)

\end{align}

</math>

,where the 3rd step can be proved by using a [[Taylor series]] expansion, and using the orthogonality of the states <math>|\text{g}\rangle</math> and <math>|\text{e}\rangle</math>. Note that a multiplication by an overall phase of <math>e^{i \omega_0 t/2}</math> on a unitary operator does not affect the underlying physics, so in the further usages of <math>U</math> we will neglect it. Applying <math>U</math> gives:

where a dot denotes the [[time derivative]]. This shows that the new Hamiltonian is given by

: <math>\begin{align}

\bar H_{H1,I} &=i\hbar\dot{equiv U} H_1 U^\dagger+UHU^ \\dagger

&=i -\hbar\left(\fracOmega e^{-i\omega_Lt} + \tilde{\hbarOmega}e^{i\omega_Lt}UH_0\right)Ue^{i\dagger+U(H_0+H_I)U^omega_0t}|\daggertext{e}\rangle\langle\text{g}|

-\hbar\left(\tilde{\Omega}^* e^{-i\omega_Lt} + \Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|e^{-i\omega_0t} \\

=UH_IU^\dagger \\

&= -\hbar\left(\Omega e^{-i\omega_gt|Delta \text{gomega t} + \rangle\langle\texttilde{g\Omega}|e^{i(\omega_L + i\omega_etomega_0)t}\right)|\text{e}\rangle\langle\text{eg}|}\left(

-\hbar\left(\tilde{\Omega }^*e^{-i(\omega_Ltomega_L + \omega_0)t} +\tilde{ \Omega}^* e^{i\omega_LtDelta \omega t}\right)|\text{eg}\rangle\langle\text{ge}|\ .

+\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|\right)

e^{-i\omega_gt|\text{g}\rangle\langle\text{g}| - i\omega_et|\text{e}\rangle\langle\text{e}|}\\

&=-e^{i\,0\,t|\text{g}\rangle\langle\text{g}| + i\omega_0t|\text{e}\rangle\langle\text{e}|}\left(

\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|

+\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|\right)

e^{-i\,0\,t|\text{g}\rangle\langle\text{g}| - i\omega_0t|\text{e}\rangle\langle\text{e}|}\\

&=-e^{i\omega_0t|\text{e}\rangle\langle\text{e}|}\left(

\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|

+\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|\right)

e^{-i\omega_0t|\text{e}\rangle\langle\text{e}|}\\

&=-\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)e^{i\omega_0t}|\text{e}\rangle\langle\text{g}|

-\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|e^{-i\omega_0t} \\

&=-\hbar\left(\Omega e^{-i\Delta t}+\tilde{\Omega}e^{i(\omega_L+\omega_0)t}\right)|\text{e}\rangle\langle\text{g}|

-\hbar\left(\tilde{\Omega}^*e^{-i(\omega_L+\omega_0)t}+\Omega^*e^{i\Delta t}\right)|\text{g}\rangle\langle\text{e}|\ .

\end{align}</math>

Now we apply the RWA by eliminating the counter-rotating terms as explained in the previous section:

The penultimate equality can be easily seen from the [[series expansion]] of the exponential map and the fact that

<math>\langle\text{i}|\text{j}\rangle=\delta_\text{ij}</math> for i and j each equal to e or g (and <math>\delta_\text{ij}</math> the [[Kronecker delta]]).

: <math>

The final step is to transform the approximate Hamiltonian back to the Schrödinger picture. The first line of the previous calculation shows that

H_{1,I}^{\text{RWA}} = -\hbar\Omega e^{-i\Delta\omega t}|\text{e}\rangle\langle\text{g}| + -\hbar\Omega^* e^{i \Delta\omega t}|\text{g}\rangle\langle\text{e}|

<math>\bar{H}=UH_IU^\dagger</math>, so in the same manner as the last calculation,

</math>

Finally, we transform the approximate Hamiltonian <math>H_{1,I}^{\text{RWA}}</math> back to the Schrödinger picture:

: <math>\begin{align}

H_{I, H_1^\text{RWA}} &= U^\dagger\bar H_{H1,I}_^{\text{RWA}} U \\

&= -\hbar\Omega e^{-i\omega_0tDelta \omega t}e^{-i\omega_0 t}|\text{e}\rangle\langle\text{eg}|}

\left( -\hbar\Omega^* e^{-i\Delta \omega t}|\text{eg}\rangle\langle\text{ge}|e^{i\omega_0t} \\

&= -\hbar\Omega^* e^{-i\Delta tomega_Lt}|\text{ge}\rangle\langle\text{eg}|\right)

-\hbar\Omega^* e^{i\omega_0tomega_Lt}|\text{eg}\rangle\langle\text{e}|} \\.

&=-\hbar\Omega e^{-i\Delta t}e^{-i\omega_0t}|\text{e}\rangle\langle\text{g}|

-\hbar\Omega^*e^{i\Delta t}|\text{g}\rangle\langle\text{e}|e^{i\omega_0t} \\

&=-\hbar\Omega e^{-i\omega_Lt}|\text{e}\rangle\langle\text{g}|

-\hbar\Omega^*e^{i\omega_Lt}|\text{g}\rangle\langle\text{e}|.

\end{align}</math>

The atomic Hamiltonian was unaffected by the approximation, so the total Hamiltonian in the Schrödinger picture under the rotating wave approximation is

:<math>

H_ H^\text{RWA} = H_0 +H_{I, H_1^\text{RWA}} =
\frac{\hbar\omega_0}{2}|\text{e}\rangle\langle\text{e}| -

- \frac{\hbar\Omega e^omega_0}{-i\omega_Lt2}|\text{eg}\rangle\langle\text{g}| -

- \hbar\Omega^* e^{-i\omega_Lt}|\text{ge}\rangle\langle\text{eg}|. -

\hbar\Omega^*e^{i\omega_Lt}|\text{g}\rangle\langle\text{e}|.

</math>

==References==

{{Reflist}}

{{DEFAULTSORT:Rotating Wave Approximation}}

[[Category:Atomic, molecular, and optical physics]]

[[Category:Chemical physics]]